A recent rumor suggests that Sony is in negotiations with 2K Games to make BioShock 4 a PlayStation exclusive. This is believed to be a limited-time deal, as Sony does not own the developer. It is also unlikely that 2K Games would forgo all the potential profits from releasing on other platforms. Numerous details about the development of the next installment have been leaking since 2019. An official reveal could happen at E3.
As per the latest episode of Xbox Era podcast, co-creator and host Nick “Shpeshal Ed” Baker claims to have heard from a number of sources that Sony is seeking to establish the sequel as a PlayStation 5 exclusive. As a third-party IP, and therefore one that it has no ownership of, the Japanese tech giant would need to hash out a deal with 2K in order to make console exclusivity worth the loss of potential sales on Xbox Series X|S.
Sources: GFR, We Got This Covered